Large-Scale Global Alignments Multiple Alignments Lecture 10, Thursday May 1, 2003.

Large-Scale Global Alignments

Multiple Alignments

Lecture 10, Thursday May 1, 2003


Rapid global alignment

Genomic regions of interest contain ordered islands of similarity

– E.g. genes

1. Find local alignments2. Chain an optimal subset of them


Suffix Trees

• Suffix trees are a method to find all maximal matches between two strings (and much more)

Example: x = dabdac

d a b d a c

ca

bd

acc

cca

db

1

4

25

63


Application: Find all Matches Between x and y

1. Build suffix tree for x, mark nodes with x

2. Insert y in suffix tree, mark all nodes y “passes from” with y

– The path label of every node marked both 0 and 1, is a common substring


Example of Suffix Tree Construction for x, y

1

x = d a b d a $y = a b a d a $

d a b d a $1. Construct tree for x

a

bd

a$2

$a

db

3

$

4

$

5

$6

xx

x

6. Insert a $

5

6

6. Insert $

4. Insert a d a $

da$

3

5. Insert d a $

y

4

2. Insert a b a d a $

a

y

da

$

1

y

yx

3. Insert b a d a $ ady

2

a$

x


Application: Online Search of Strings on a Database

Say a database D = { s1, s2, …sn }(eg. proteins)

Question: given new string x, find all matches of x to database

1. Build suffix tree for {s1,…, sn}2. All new queries x take O( |x| ) time

(somewhat like BLAST)


Application: Common Substrings of k Strings

• Say we want to find the longest common substring of s1, s2, …sn

1. Build suffix tree for s1,…, sn

2. All nodes labeled {si1, …, sik} represent a match between si1, …, sik

Methods to CHAIN Local Alignments

Sparse Dynamic ProgrammingO(N log N)


The Problem: Find a Chain of Local Alignments

(x,y) (x’,y’)

requires

x < x’y < y’

Each local alignment has a weight

FIND the chain with highest total weight


Quadratic Time Solution

• Build Directed Acyclic Graph (DAG):– Nodes: local alignments (xa,xb) (ya,yb) Directed Edges:

any two local alignments that can be chained

• Each local alignment is a node vi with alignment score si


Quadratic Time Solution (cont’d)

Dynamic programming:

Initialization:Find each node va s.t. there is no edge (u,v0)

Set score of V(a) to be sa

Iteration:For each vi, optimal path ending in vi has total score:

V(i) = max ( weight(vj, vi) + V(j) )Termination:

Optimal global chain: j = argmax ( V(j) ); trace chain from vj

Worst case time: quadratic


Faster Solution: Sparse DP

• 1,…, N: rectangles

• (hj, lj): y-coordinates of rectangle j

• w(j): weight of rectangle j

• V(j): optimal score of chain ending in j

• L: list of triplets (lj, V(j), j)

L is sorted by ljy

h

l


Sparse DP

Go through rectangle x-coordinates, from lowest to highest:

1. When on the leftmost end of i:a. j: rectangle in L, with largest lj < hi

b. V(i) = w(i) + V(j)

2. When on the rightmost end of i:a. j: rectangle in L, with largest lj lib. If V(i) > V(j):

i. INSERT (li, V(i), i) in L

ii. REMOVE all (lk, V(k), k) with V(k) V(i) & lk li

x

y


Example

x

y

1: 5

3: 3

2: 6

4: 45: 2

2

56

91011

1214

1516


Time Analysis

1. Sorting the x-coords takes O(N log N)

2. Going through x-coords: N steps

3. Each of N steps requires O(log N) time:• Searching L takes log N• Inserting to L takes log N• All deletions are consecutive, so log N


Putting it All Together:Fast Global Alignment Algorithms

1. FIND local alignments2. CHAIN local alignments

FIND CHAINGLASS: k-mers hierarchical DPMumMer: Suffix Tree Sparse DPAvid: Suffix Tree hierarchical DP

LAGAN CHAOS Sparse DP

LAGAN: Pairwise Alignment

1. FIND local alignments

2. CHAIN local alignments

3. DP restricted around chain


LAGAN

1. Find local alignments

2. Chain -O(NlogN) L.I.S.

3. Restricted DP


LAGAN: recursive call

• What if a box is too large?– Recursive application of LAGAN,

more sensitive word search


3. LAGAN: memory

“necks” have tiny tracebacks

…only store tracebacks

Multiple Sequence Multiple Sequence AlignmentsAlignments




Overview

• Definition

• Scoring Schemes

• Algorithms

Multiple Sequence Alignments

Definition and Motivation


Definition

• Given N sequences x1, x2,…, xN:– Insert gaps (-) in each sequence xi, such that

• All sequences have the same length L• Score of the global map is maximum

• Pairwise alignment: a hypothesis on the evolutionary relationship between the letters of two sequences

• Same for a multiple alignment!


Motivation

• A faint similarity between two sequences becomes very significant if present in many

• Protein domains

• Motifs responsible for gene regulation


Another way to view multiple alignments

• Pairwise alignment:

Good alignment Evolutionary relationship

• Multiple alignment:

Something in common Sequence in common

“inverse” to pairwise alignment


Scoring Function


Scoring Function

• Ideally:– Find alignment that maximizes probability that sequences

evolved from common ancestor

x

yz

w

v

?


Scoring Function (cont’d)

• Unfortunately: too many parameters

• Compromises:

– Ignore phylogenetic tree

– Statistically independent columns:

S(m) = G(m) + i S(mi)

m: alignment matrixG: function penalizing gaps


Scoring Function: Sum Of Pairs

Definition: Induced pairwise alignmentA pairwise alignment induced by the multiple alignment

Example:

x: AC-GCGG-C y: AC-GC-GAG z: GCCGC-GAG

Induces:

x: ACGCGG-C; x: AC-GCGG-C; y: AC-GCGAGy: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG


Sum Of Pairs (cont’d)

• The sum-of-pairs score of an alignment is the sum of the scores of all induced pairwise alignments

S(m) = k<l s(mk, ml)

s(mk, ml): score of induced alignment (k,l)


Sum Of Pairs (cont’d)

• Heuristic way to incorporate evolution tree:

Human

Mouse

Chicken• Weighted SOP:

S(m) = k<l wkl s(mk, ml)

wkl: weight decreasing with distance

Duck


Consensus

-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC---TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGACCAG-CTATCAC--GACCGC----TCGATTTGCTCGAC

CAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC

• Find optimal consensus string m* to maximize

S(m) = i s(m*, mi)

s(mk, ml): score of pairwise alignment (k,l)


Algorithms


Multidimensional Dynamic Programming

Generalization of Needleman-Wunsh:

S(m) = i S(mi)

(sum of column scores)

F(i1,i2,…,iN) = max(all neighbors of cube)(F(nbr)+S(nbr))


• Example: in 3D (three sequences):

• 7 neighbors/cell

F(i,j,k) = max{ F(i-1,j-1,k-1)+S(xi, xj, xk),F(i-1,j-1,k )+S(xi, xj, - ),F(i-1,j ,k-1)+S(xi, -, xk),F(i-1,j ,k )+S(xi, -, - ),F(i ,j-1,k-1)+S( -, xj, xk),F(i ,j-1,k )+S( -, xj, xk),F(i ,j ,k-1)+S( -, -, xk) }




Running Time:

1. Size of matrix: LN;

Where L = length of each sequence N = number of sequences

2. Neighbors/cell: 2N – 1

Therefore………………………… O(2N LN)

Large-Scale Global Alignments Multiple Alignments Lecture 10, Thursday May 1, 2003.

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